The wavepacket modulation can be more clearly evaluated by analysing the structure of each modulation component, which can be isolated from the total field directly in (REF ); for instance, by neglecting the terms \(\mathbf {\hat{q}_{-1}}\)  and \(\mathbf {\hat{q}_{0}}\) , the field associated with the positive modulation (\(\mathbf {\hat{q}_{+1}}\) ) is obtained. The zeroth, first positive and first negative components of the modulation are shown in figures REF (c-h) and REF (c-h). As the modulation of the wavepacket comes from the modulation wavenumbers, the zeroth component is directly comparable to PSE, possessing the same qualitative features. Comparing with the total field, it is also clear that this component is the most energetic one, defining the shape of the final solution. Overall, the shapes of the positive modulation components are very similar for both frequencies, with a single peak close to the centreline and almost zero amplitude in the subsonic region of the jet; this is consistent with the hypothesis that the effect of the shock-cell on the KH mode should be more evident close to the peaks of the shocks in the simplified model. These components also follow the growth/decay behaviour of the KH mode, which agrees with the theory developed by [1]}; as the KH mode is the only structure capable of extracting energy from the mean flow, the energy growth of the modulation wavenumbers must be chained to the growth of the unstable mode. As expected, the positive part of the modulation is associated with higher wavenumbers. The negative component of the modulation is also similar for \(St=0.4\) ; as the wavenumber of this structure is given by the difference between the wavenumber of the KH mode and the local wavenumber of the shocks, this modulation leads to structures with larger wavelength (when compared to the positive modulation). Interestingly, the negative part of the modulation for \(St=0.7\)  (figures REF (h) and REF (h)) differs from its low frequency counterpart, having a double-peak structure. Considering that the second radial order guided jet mode is propagative for this frequency and that the resulting shape of this modulation component is very similar to the cited mode [2]}, it is possible that the presence of the neutral upstream mode in the spectrum is affecting the spatial march, leading to the appearance of this mode for negative wavenumbers, instead of the modulation of the KH mode. This is also supported by the apparent switch from single- to double-peak structure at the early stations of the jet, also seen in figure REF (h); thus, even though the initial solution has a single-peak modulation, the PFE march can induce the appearance of the guided jet mode in the final wavepacket.
